Answer
$(12,6)$, value $42$
Work Step by Step
1. Graph the inequalities as shown in the figure where the solution is in the first quadrant.
2. We can identify the vertices as shown in the figure.
3. Calculate the objective function values at these vertices as $z_1=2(0)+3(0)=0, z_2=2(0)+3(12)=36, z_3=2(12)+3(6)=42, z_4=2(20)+3(0)=40$
4. Thus, the maximum happens at $(12,6)$ with a value $42$
